Let us recall that one can use an approximate expression for all these sums, which partially simplifies the gate-infidelity expressions [1]}. In any case, it is easy to realise that the total crosstalk contribution to the MS-gate infidelity, obtained by adding all of the above terms, can be expressed in terms of the infidelity \(\epsilon _{\rm MS}\)  of the MS-gate without crosstalk [2]} as follows
\(\epsilon ^{\rm ct}_{\rm MS}=&\sum _{j=1}^M\left(\frac{\Omega _j}{\Omega }\right)^{\!\!2}\left(\epsilon _{\rm MS}+\frac{\pi ^2}{4}N +2N^2\frac{t_{\rm g}}{T_2}\right)\nonumber \\&-2N^2\frac{t_{\rm g}}{T_2}.\) 
